Evidence against the polarization rotation model of piezoelectric perovskites at the morphotropic phase boundary
aa r X i v : . [ c ond - m a t . m t r l - s c i ] O c t Evidence against the polarization rotation model of piezoelectric perovskites at themorphotropic phase boundary
J. Frantti, ∗ Y. Fujioka, and R. M. Nieminen
COMP/Department of Applied Physics, Helsinki University of Technology, FI-02015-HUT, Finland (Dated: November 2, 2018)The origin of the very large piezoelectric response observed in the vicinity of the morphotropicphase boundary (MPB) in perovskite lead zirconate titanate and related systems has been underintensive studies. Polarization rotation ideas are frequently invoked to explain the piezoelectricproperties. It was recently reported that lead titanate undergoes a phase transformation sequence P mm → P m → Cm → R ¯3 c at 10 K as a function of hydrostatic pressure [M. Ahart et al.Nature Letters. , 545 (2008)]. We demonstrate that this interpretation is not correct by (i)simulating the reported diffraction patterns, and (ii) by density-functional theory computationswhich show that neither the P m , Cm nor P mm
The polarization rotation (PR) model [1, 2] has beenproposed to explain the large electromechanical couplingcoefficients observed in ferroelectric perovskites in thevicinity of the morphotropic phase boundary (MPB).The MPB region separates tetragonal and rhombohedralphases, which do not have a group-subgroup relationshipand thus no continuous transition between the phases ispossible. The most intensively studied systems are solidsolutions, prime examples being lead zirconate titanate,Pb(Zr x Ti − x )O , (PZT) and x Pb(Mg / Nb / )O -(1 − x )PbTiO (PMN-PT). The essential feature of the PRmodel is the insertion of one (or more) low-symmetryphase(s) to continuously (via group-subgroup chains)connect the tetragonal and rhombohedral phases sepa-rated by the MPB in order to continuously rotate thepolarization vector by an electric field or pressure be-tween the pseudo-cubic [001] and [111] directions alongthe (1¯10) plane. This rotation path was predicted to beaccompanied by a large electromechanical response [3].There are, however, several ambiguities related to the PRmodel (see, e.g., Ref. 4) and experimental studies inter-preted in terms of this idea. As an example, the pressureinduced phase transitions of lead titanate (PbTiO , PT)are considered below. Hydrostatic pressure induces simi-lar structural changes as are observed to occur due to thesubstitution of Ti by a larger cation, such as Zr, causingso called “chemical pressure”.At high temperatures PT undergoes a phase transi-tion between the P mm and P m ¯3 m phases [5]. At roomtemperature PT transforms to a cubic phase through asecond-order transition at 12.1 GPa[6], whereas it waspredicted through density-functional theory (DFT) com-putations that a phase transition between P mm and R c phases occurs at 9 GPa at 0 K [7]. Notably thelatter phase transition is similar to the phase transitionobserved in PZT as a function of Zr composition. In simplest terms, one expects to have three different phaseboundaries in the pressure-temperature plane of PT, sep-arating the P mm and P m ¯3 m , P mm and R c and R c and P m ¯3 m phases. A very different interpretation wasrecently given in Ref. 8, according to which the phasetransition from the P mm to R ¯3 c phase would occurvia monoclinic phases, which was further claimed to givesupport to the PR model. We demonstrate that (i) thesingle phase model is incorrect in the vicinity of the phasetransition, (ii) the monoclinic distortions reported ear-lier are not stable, (iii) summarize the arguments whichshow that the phase transition must be of first order and(iv) outline the method for determining the piezoelectricproperties in the vicinity of the phase boundary. Computational methods.
The DFT code ABINIT [9,10] was used to compute the total energies and phononfrequencies and eigenvectors [11] at different pressures.The computations were carried out within the local-density approximation and a plane wave basis. Norm-conserving pseudopotentials were generated using theOPIUM package [12]. A more detailed description ofthe computational approach is available in Ref. 7. Forthe simulation of the X-ray diffraction patterns the Pow-der Cell program was used [13]. The lattice parameterswere adapted from Ref. 8. The asymmetric unit wasnot given in Ref. 8, and thus the atomic positions wereestimated using the values found from the DFT compu-tations, which are close to the values estimated from ourhigh-pressure neutron powder diffraction experiments atfew GPa pressures [14].
Notes on the X-ray diffraction and Raman scatteringanalysis.
According to Ref. 8, PT undergoes a phasetransformation sequence P mm → P m → Cm → R ¯3 c at 10 K as a function of hydrostatic pressure. We showthat the X-ray diffraction (XRD) pattern collected at13.2 GPa [8] is not consistent with the reported P m sym-metry by simulating the corresponding pattern. Fig. 1shows that the reflection positions and intensities signifi-cantly deviate from the experimental ones and also fromthe fits (shown by black continuous lines). It is worth tonote that in the case of PT the pseudo-cubic 110 reflec-tions have the strongest XRD intensities. The 13.2 GPaXRD pattern shown in Fig. 1 more likely corresponds toa two-phase diffraction pattern. This is seen by studyingthe intensities of the 100 and 001 reflections: for tetrag-onal and pseudo-tetragonal structures the intensity ratioshould roughly be 2:1 (as is seen from the diffraction pat-tern collected at 8.4 GPa, Fig. 1), whereas it is roughly0.9:1 for the 13.2 GPa data.It was stated that the Raman scattering data reflectthe monoclinic M C ( P m phase) to monoclinic M A ( Cm )and the monoclinic M A to rhombohedral phase transi-tions [8]. We find this assignment questionable, since thephonon symmetries, central for the phase transition stud-ies, were not addressed. For example, the B -symmetrynormal mode in the P mm phase breaks the fourfoldsymmetry [16], whereas the A symmetry modes preserveit. The spectral features below 100 cm − include severalpeaks from the A symmetry modes alone, due to thestrong anharmonicity of the A (1TO) mode [17, 18], inaddition to the E -symmetry modes and Rayleigh scatter-ing (which dominates the region close to the laser line,as was noted in Ref. 6). It was rather recently that the A (1TO) mode was identified in PT[17, 18]: many earlierassignments dismissed this mode since the line shape wasvery asymmetric and turned out to be consisted of manysubpeaks. In practice this means that, in the vicinity ofthe phase transition, it is hard to identify the number ofmodes at the low-frequency region, not to mention thedifficulty of identifying their symmetries from the spec-tra collected without proper polarization measurements.This, in turn, prevents space group assignments. DFT studies.
DFT computations predict that PT un-dergoes a phase transition from the P mm phase to the R c phase at around 9 GPa [7]. In contrast, a phasetransition sequence P mm → Cm → R m → P m ¯3 m (phase transitions at 10, 12 and 22 GPa, respectively)was found in Ref. 19. The high-pressure end of this tran-sition was more recently modified to form the sequence R m → R c → R ¯3 c → R c with phase transitions occur-ring at 18, 20 and 60 GPa, respectively [8]. In addition tothe phases listed in Ref. 7, we carried out similar compu-tations for the P m and
P mm R c phase were computed at9, 10 and 15 GPa pressures at the Brillouin zone centerand boundary points.The main outcomes of our present and earlier com-putations are: (i) the R m phase is not stable (octahe-dral tilting makes R c phase favorable above 9 GPa), (ii)above 9 GPa tetragonal ( P mm and I cm ), orthorhom-bic ( Cmm
P mm
2) and monoclinic (
P m and Cm )phases were revealed to be unstable by the Brillouin zone boundary modes and higher enthalpy values, (iii)no support for an intermediate phase was found, and(iv) no phonon instabilities were observed in the R c phase. In contrast, one of the Brillouin zone corner point L = ( πa πa πa ) modes of the R m phase was unstable at9 GPa pressure. The mode involved only oxygen dis-placements (this was the only mode which was found tobe unstable: all modes at the (000), (00 πa ) and ( πa πa R c phase. This is due to the fact that octahedral tiltingallows a more efficient compression [7, 20, 21].We note that since the R m phase is not stable, itis somewhat hypotetical to consider the instability ofan unstable phase. A more rigorous treatment, start-ing from the P mm phase, is given in Ref. 7, with thesame outcome. Thus, the energetically favorable phasewas obtained by allowing the crystal to relax accordingto the normal mode displacements of the unstable modesseen in the P mm phase. Thus the transition between P mm and R c phases is characterized by two-phase co-existence, in an analogous way to the phase transitionsseen in PZT as a function of composition. This is animportant prediction as it in turn suggests that the two-phase co-existence has a crucial role for the piezoelectricproperties near the phase transition pressures in PT, ina similar way as was demonstrated in Ref. 22 for PZT inthe vicinity of the MPB. Symmetry considerations.
Group-theoretical analysisindicates that, although the phase transition betweenmonoclinic and tetragonal phases can be continuous, thetransition between rhombohedral and monoclinic phasesmust be of first order [23]. Thus, even if one would havea monoclinic phase, it would not make the transforma-tion path continuous. First-order transitions are oftencharacterized by the two-phase co-existence, one phasebeing metastable over a finite temperature or pressurerange. This is consistent with the experimentally knownfeatures of PZT according to which there is two-phase co-existence [21, 24, 25]. Neutron and X-ray powder diffrac-tion studies revealed that the polarization vector in themonoclinic Cm phase is very close to the pseudo-cubic[001] direction, and hardly rotates from that direction[21, 25], in contrast to what one anticipates from the PRmodel. Thus the polarization vector changes discontin-uously when the transition from the pseudo-tetragonalmonoclinic to the rhombohedral phase occurs. As Li etal. noted, “the availability of multiple phases at the MPBmakes it possible for the polarization to thread throughthe ceramic”[22]. How to model the piezoelectric response?
The piezo-electric response can be divided to extrinsic and intrin-sic contributions. The latter is due to the changes in
FIG. 1: X-ray diffraction data collected on PT at 10 K. The figure is adapted from Ref. 8. The green and blue lines (middlepanels) were added by us. The green line shows the simulated
P m pattern using the lattice parameters given in Ref. 8. Themodel where the a and b axes are switched (blue line) does not improve the fit [15]. Neither of the one-phase P m structuremodel fits the peak (black lines) positions and intensities (e.g., the reflection labelled as (¯110) is not modelled, and cannot beexplained by preferred orientation). electron densities as a response to an applied field orstress and can be computed through standard density-functional theory methods. The extrinsic part is signifi-cantly more challenging, as it involves domain wall mo-tions and changes in the phase fractions in the vicinity ofthe phase-boundary (e.g., between tetragonal and rhom-bohedral phases). In the case of poled ceramics one firstcomputes the necessary angular averages of the piezo-electric constants and takes their dependence on temper-ature, composition or stress into account. This depen-dence is notable in the vicinity of the phase transition.For intrinsic contribution such a computation is ratherstraightforward. However, the description of domain wallmotion due to an applied electric field or stress for differ-ent composition or at different temperatures is nontrivialtask.In conclusion, evidence against the applicability of thepolarization rotation model to perovskites is strong. In-stead, the currently known best piezoelectric perovskitesposses a so-called morphotropic phase boundary at which a first-order phase transition between rhombohedral andtetragonal (or pseudo-tetragonal) phases takes place.For the electromechanical properties it is important tonote that this transition exhibits two-phase co-existence.Structural factors responsible for the stabilization of therhombohedral phase, either at large hydrostatic pressuresor large chemical pressures (as occurs in Pb(Zr x Ti − x )O with increasing x ) were addressed.Y. F. is grateful for the Finnish Cultural Foundationand Tekniikan Edist¨amis¨a¨ati¨o foundation for financialsupport. This project was supported by the Academy ofFinland (Project Nos 207071 and 207501 and the Centerof Excellence Program 2006-2011). Finnish IT Center forScience (CSC) is acknowledged for providing computingresources. ∗ johannes.frantti@hut.fi FIG. 2: . The unstable normal mode of the R m phase at the L = ( πa πa πa ) point involves only oxygen ions. Two rhombohe-dral unit cells are shown: it is seen that the two octahedra aretilted about the threefold axes clockwise and anticlockwise.The condensation of this mode corresponds to the phase tran-sition R m → R c . All the modes had positive frequencieswhen the R c phase was used. The bold line is the threefoldrotation axis.[1] H. Fu and R. E. Cohen, Nature. , 281 (2000).[2] R. Guo, L. E. Cross, S.-E. Park, B. Noheda, D. E. Cox,and G. Shirane, Phys. Rev. Lett. , 5423 (2000).[3] R. E. Cohen, Nature. , 941 (2006).[4] E. H. Kisi and J. S. Forrester, J. Phys. Condens. Matter. , 1 (2008).[5] A. Sani, M. Hanfland, and D. Levy, J. Solid State Chem. , 446 (2002).[6] J. A. Sanjurjo, E. L´opez-Cruz, and G. Burns, Phys. Rev.B. , 7260 (1983).[7] J. Frantti, Y. Fujioka, and R. M. Nieminen, J. Phys.Chem. B. , 4287 (2007).[8] M. Ahart, M. Somayazulu, R. E. Cohen, G. P. Dera,H. Mao, J. Russell, J. Hemley, Y. Ren, P. Liermann, and Z. Wu, Nature , 545 (2008).[9] X. Gonze, J.-M. Beuken, R. Caracas, F. Detraux,M. Fuchs, G.-M. Rignanese, L. Sindic, M. Verstraete,G. Zerah, F. Jollet, et al., Comput. Mater. Sci. , 478(2002).[10] X. Gonze, G.-M. Rignanese, M. Verstraete, J.-M.Beuken, Y. Pouillon, R. Caracas, F. Jollet, M. Torrent,G. Zerah, M. Mikami, et al., Zeit. Kristallogr. , 558(2005).[11] X. Gonze, Phys. Rev. B. , 10337 (1997).[12] A. M. Rappe, K. M. Rabe, E. Kaxiras, , and J. D.Joannopoulos, Phys. Rev. B. , 1227 (1990).[13] W. Krans and G. Nolze (1996).[14] J. Frantti, Y. Fujioka, J. Zhang, S. Vogel, S. Wang, andR. M. Nieminen, Unpublished.[15] According to the authors of Ref. 8, a and b were inadver-tently switched for P m . Information obtained throughNature Editorial Office.[16] J. Frantti, J. Lappalainen, V. Lantto, S. Nishio, andM. Kakihana, J. J. Appl. Phys. , 5679 (1999).[17] C. M. Foster, M. Grimsditch, Z. Li, and V. G. Karpov,Phys. Rev. Lett. , 1258 (1993).[18] C. M. Foster, Z. Li, M. Grimsditch, S.-K. Chan, and D. J.Lam, Phys. Rev. B. , 10160 (1993).[19] Z. Wu and R. E. Cohen, Phys. Rev. Lett. , 037601(2005).[20] N. W. Thomas and A. Beitollahi, Acta Cryst. B. , 549(1994).[21] J. Frantti, S. Ivanov, S. Eriksson, H. Rundl¨of, V. Lantto,J. Lappalainen, and M. Kakihana, Phys. Rev. B. ,064108 (2002).[22] J. Y. Li, R. C. Rogan, E. ¨Ust¨undag, and K. Bhat-tacharya, Nature Mat. , 4 (2005).[23] I. A. Sergienko, Y. M. Gufan, and S. Urazhdin, Phys.Rev. B. , 144104 (2002).[24] D. E. Cox, B. Noheda, and G. Shirane, Phys. Rev. B. ,134110 (2005).[25] J. Frantti, S. Eriksson, S. Hull, V. Lantto, H. Rundl¨of,and M. Kakihana, J. Phys. Condens. Matter.15